passing through it, where X, Y, Z are current co-ordinates, and call
this association a surface-element; and if two consecutive elements of
which the point(x + dx, y + dy, z + dz) of one lies on the plane of the
other, for which, that is, the condition dz = pdx + qdy is satisfied, be
said to be _connected,_ and an infinity of connected elements following
one another continuously be called a _connectivity_, then our statement
is that a connectivity consists of not more than [oo]^2 elements, the
whole number of elements (x, y, z, p, q) that are possible being called
[oo]^5. The solution of an equation F(x, y, z, dz/dx, dz/dy) = 0 is then
to be understood to mean finding in all possible ways, from the [oo]^4
elements (x, y, z, p, q) which satisfy F(x, y, z, p, q) = 0 a set of
[oo]^2 elements forming a connectivity; or, more analytically, finding in
all possible ways two relations G = 0, H = 0 connecting x, y, z, p, q
and independent of F = 0, so that the three relations together may
involve
dz = pdx + qdy.
Such a set of three relations may, for example, be of the form z =
[psi](x, y), p = d[psi]/dx, q = d[psi]/dy; but it may also, as another
case, involve two relations z = [psi](y), x = [psi]1(y) connecting x, y,
z, the third relation being
[psi]'(y) = p[psi]'1(y) + q,
the connectivity consisting in that case, geometrically, of a curve in
space taken with [oo]^1 of its tangent planes; or, finally, a
connectivity is constituted by a fixed point and all the planes passing
through that point. This generalized view of the meaning of a solution
of F = 0 is of advantage, moreover, in view of anomalies otherwise
arising from special forms of the equation itself. For instance, we may
include the case, sometimes arising when the equation to be solved is
obtained by transformation from another equation, in which F does not
contain either p or q. Then the equation has [oo]^2 solutions, each
consisting of an arbitrary point of the surface F = 0 and all the [oo]^2
planes passing through this point; it also has [oo]^2 solutions, each
consisting of a curve drawn on the surface F = 0 and all the tangent
planes of this curve, the whole consisting of [oo]^2 elements; finally,
it has also an isolated (or singular) solution consisting of the points
of the surface, each associated with the tangent plane of the surface
thereat, also [oo]^2 elements in all. Or again, a linear equation F = Pp
+ Qq - R = 0, wherein P, Q, R are functions of x, y, z only, has [oo]^2
solutions, each consisting of one of the curves defined by
dx/P = dy/Q = dz/R
taken with all the tangent planes of this curve; and the same equation
has [oo]^2 solutions, each consisting of the points of a surface
containing [oo]^1 of these curves and the tangent planes of this surface.
And for the case of n variables there is similarly the possibility of n
+ 1 kinds of solution of an equation F(x1, ... xn, z, p1, ... pn) = 0;
these can, however, by a simple contact transformation be reduced to one
kind, in which there is only one relation z' = [psi](x'1, ... x'n)
connecting the new variables x'1, ... x'n, z' (see under PFAFFIAN
EXPRESSIONS); just as in the case of the solution
z = [psi](y), x = [psi]1(y), [psi]'(y) = p[psi]'1(y) + q
of the equation Pp + Qq = R the transformation z' = z - px, x' = p, p' =
-x, y' = y, q' = q gives the solution
z' = [psi](y') + x'[psi]1(y'), p' = dz'/dx', q' = dz'/dy'
of the transformed equation. These explanations take no account of the
possibility of p and q being infinite; this can be dealt with by writing
p = -u/w, q = -v/w, and considering homogeneous equations in u, v, w,
with udx + vdy + wdz = 0 as the differential relation necessary for a
connectivity; in practice we use the ideas associated with such a
procedure more often without the appropriate notation.
Order of the ideas.
In utilizing these general notions we shall first consider the theory of
characteristic chains, initiated by Cauchy, which shows well the nature
of the relations implied by the given differential equation; the
alternative ways of carrying out the necessary integrations are
suggested by considering the method of Jacobi and Mayer, while a good
summary is obtained by the formulation in terms of a Pfaffian
expression.
Characteristic chains.
Consider a solution of F = 0 expressed by the three independent
equations F = 0, G = 0, H = 0. If it be a solution in which there is
more than one relation connecting x, y, z, let new variables x', y',
z', p', q' be introduced, as before explained under PFAFFIAN
EXPRESSIONS, in which z' is of the form
z' = z - p1x1 - ... - p_s x_s (s = 1 or 2),
so that the solution becomes of a form z' = [psi](x'y'), p' =
d[psi]/dx', q' = d[psi]/dy', which then will identically satisfy the
transformed equations F' = 0, G' = 0, H' = 0. The equation F' = 0, if
x', y', z' be regarded as fixed, states that the plane Z - z' = p'(X -
x') + q'(Y - y') is tangent to a certain cone whose vertex is (x', y',
z'), the consecutive point (x' + dx', y' + dy', z' + dz') of the
generator of contact being such that
/dF' /dF' / / dF' dF'\
dx'/ -- = dy'/ -- = dz'/ ( p'--- + q' --- ).
/ dp' / dq' / \ dp' dq'/
Passing in this direction on the surface z' = [psi](x', y') the
tangent plane of the surface at this consecutive point is (p' + dp',
q' + dq'), where, since F'(x', y', [psi], d[psi]/dx', d[psi]/dy') = 0
is identical, we have dx' (dF'/dx' + p'dF'/dz') + dp'dF'/dp' = 0. Thus
the equations, which we shall call the characteristic equations,
/dF' /dF' // dF' dF'\ // dF' dF'\
dx'/ --- = dy'/ --- = dz'/( p' --- + q'--- ) = dp'/( - --- - p'--- )
/ dp' / dq' / \ dp' dq'/ / \ dx' dz'/
// dF' dF'\
= dq'/( - --- - q'--- )
/ \ dy' dz'/
are satisfied along a connectivity of [oo]^1 elements consisting of a
curve on z' = [psi](x', y') and the tangent planes of the surface
along this curve. The equation F' = 0, when p', q' are fixed,
represents a curve in the plane Z - z' = p'(X - x') + q'(Y - y')
passing through (x', y', z'); if (x' + [delta]x', y' + [delta]y', z' +
[delta]z') be a consecutive point of this curve, we find at once
/dF' dF'\ /dF' dF'\
[delta]x'( --- + p'--- ) + [delta]y'( --- + q'--- ) = 0;
\dx' dz'/ \dy' dz'/
thus the equations above give [delta]x'dp' + [delta]y'dq' = 0, or the
tangent line of the plane curve, is, on the surface z' = [psi](x',
y'), in a direction conjugate to that of the generator of the cone.
Putting each of the fractions in the characteristic equations equal to
dt, the equations enable us, starting from an arbitrary element x'0,
y'0, z'0, p'0, q'0, about which all the quantities F', dF'/dp', &c.,
occurring in the denominators, are developable, to define, from the
differential equation F' = 0 alone, a connectivity of [oo]^1 elements,
which we call a _characteristic chain_; and it is remarkable that when
we transform again to the original variables (x, y, z, p, q), the form
of the differential equations for the chain is unaltered, so that they
can be written down at once from the equation F = 0. Thus we have
proved that the characteristic chain starting from any ordinary
element of any integral of this equation F = 0 consists only of
elements belonging to this integral. For instance, if the equation do
not contain p, q, the characteristic chain, starting from an arbitrary
plane through an arbitrary point of the surface F = 0, consists of a
pencil of planes whose axis is a tangent line of the surface F = 0. Or
if F = 0 be of the form Pp + Qq = R, the chain consists of a curve
satisfying dx/P = dy/Q = dz/R and a single infinity of tangent planes
of this curve, determined by the tangent plane chosen at the initial
point. In all cases there are [oo]^3 characteristic chains, whose
aggregate may therefore be expected to exhaust the [oo]^4 elements
satisfying F = 0.
Complete integral constructed with characteristic chains.
Consider, in fact, a single infinity of connected elements each
satisfying F = 0, say a chain connectivity T, consisting of elements
specified by x0, y0, z0, p0, q0, which we suppose expressed as
functions of a parameter u, so that